Chapter 1
.
Graphical
Displays of
Univariate Data
Topic 2 covers sorting data and constructing Stemplots and
Dotplots, Topic 3 Histograms, and Topic 4 Frequency
Plots. (Note: Boxplots are a graphical display but will be
covered in Chapter 2, Topic 7.)
Topic 2—Stemplots, Dotplots and Sorting Data
Example: Tall building heights in Philadelphia, PA.
Save the following data, in order, in a list called yrphil in
folder BLDTALL. Arrange the Stats/List Editor in folder
BLDTALL as shown in screen 4 on the next page (phily,
yrphil, list1, list2).
Year the 24 Tallest Buildings in Philadelphia, PA were Completed
1901
1927
1929
1930
1930
1930
1932
1969
1970
1973
1973
1973
1974
1982
1983
1987
1988
1989
1990
1990
1990
1991
1992
XXXX
(Source: Reprinted with permission from the World Almanac and Book of Facts 2000. © 2000 World Almanac Education
Group, Inc. All rights reserved.)
In the table, XXXX represents a missing value.
The TI-89 does not have stemplots or dotplots built in, but
these are easy to construct by hand if the data is in order.
© 2001 TEXAS INSTRUMENTS INCORPORATED
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ADVANCED PLACEMENT STATISTICS WITH THE TI-89
Forming a New List
1.
Highlight list1, press 2 /, and type yrphil. Press
¸ (screen 1).
(1)
2.
Type the 24 years by rows (given in the table) into the
yrphil list (screen 2).
(2)
3.
Highlight yrphil, press 2 /, and type or paste phily
in the input line. Press ¸ (screen 3).
(3)
Putting Data in Order
To compare the data entered in list phily, you can sort the
list in ascending order. To retain the relationship between
phily and yrphil, however, you should make a copy of the list
before you sort. You realize the building built in 1901 was
585 feet and not 400 feet, as shown in the first row of list1
(screen 7).
1.
Make a copy of list phily in list1:
a.
Highlight the name list1 (screen 4).
b. Paste or type phily in the input line.
(4)
© 2001 TEXAS INSTRUMENTS INCORPORATED
CHAPTER 1: GRAPHICAL DISPLAYS OF UNIVARIATE DATA
c.
23
Press ¸ (screen 5).
(5)
2.
Sort list1 in ascending order (smallest to largest):
a.
Press … List.
b. Select 2:Ops and then 1:Sort List (screen 6).
(6)
3.
If list1 or any element of list1 was highlighted before
step 2, then list1 is in the List: field of the Sort List dialog
box. If not, paste or type it.
4.
Press ¸ to execute the sort (screen 7). Notice that
the smallest building in list phily is 400 feet.
(7)
Keeping the Relationship Between Two Lists When Sorting
Make a copy of list phily in list2 and list yrphil in list3. (See
step 1 above.) If list2 is a copy of phily and list3 is a copy of
yrphil, you could sort on List: list2, list3 (screen 8).
1.
Press … List.
2.
Select 2:Ops and then 1:Sort List.
3.
In the entry line, type list2, list3 (screen 8).
4.
Press ¸ ¸.
(8)
© 2001 TEXAS INSTRUMENTS INCORPORATED
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ADVANCED PLACEMENT STATISTICS WITH THE TI-89
The first two outputs of list2 after sorting are 400 feet, but
the correct values of years built is also in list3 (1929 and
1982). (See screen 9.)
(9)
Creating a Stemplot (Stem-and-Leaf Plots)
Make a sorted list for phily data as in list1 above.
The first six values of list1 measured in ten-foot units and
rounded to the nearest ten feet are 40, 40, 41, 41, 42, and 42,
with a stem of 4 and leaves 0, 0, 1, 1, 2, and 2 for 4:001122.
Press 2 D for the next six values and continue to get data
for the complete stemplot.
.
4:001122445889999
5:0779
(City Hall)
6:
7:049
8:5
9:5
(One Liberty Place)
Stemplot of Tall Buildings in Philadelphia
© 2001 TEXAS INSTRUMENTS INCORPORATED
CHAPTER 1: GRAPHICAL DISPLAYS OF UNIVARIATE DATA
25
Creating a Dotplot
For each value in list2, place one dot above its height on the
number line as shown (screen 4). If there are repeated
values (or values close together) stack them. A method of
creating a dotplot on the TI-89 will be explained in Topic 9,
screens 13 and 14.
One Liberty Place
City Hall
Phily Dotplot
400
500
600
700
800
900
1000
Height of Buildings (ft.)
While all of the buildings are tall (400 feet or taller), two are
more than twice as tall as the smallest of 400 feet with the
tallest at 945 feet, a 545-foot difference or spread.
The distribution is skewed to the higher values with 67%
(16 of 24) of the heights from 400 to 500 feet and only five
values (21%) are 700 feet or taller.
City Hall stands in a little cluster of three buildings that
stand out more in the dotplot than in the stemplot. One
Liberty Place is the tallest building and is a bit of an outlier,
with the nearest building being almost 100 feet shorter.
There is another cluster of buildings about 490 feet tall, with
a small gap between them and the smaller buildings. There
is a larger gap between them and the taller “City Hall three.”
Finally, there is a 115-foot gap between City Hall and the
700-foot building which could be considered part of a widely
spread cluster of four buildings.
Note: One Liberty Place (945 feet)
was completed in 1987. Until that
time, by agreement, no building
was higher than the hat on the
William Penn statue atop City Hall
(585 feet).
Outliers are values that are far
away from the rest of the data and
that may have a special story.
© 2001 TEXAS INSTRUMENTS INCORPORATED
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ADVANCED PLACEMENT STATISTICS WITH THE TI-89
Topic 3—Histograms
Constructing Histograms and Frequency Tables from a
Data List
Example: Use the phily list of building heights from Topic 1
with n = 24, minX = 400 feet, and maxX = 945 feet (Topic 1,
screens 37 and 38).
The following approximation can be used to determine the
number of cells to use for a given list size.
n (number of
elements)
<25
25
50
100
200
400
800
1600
...
Double
previous value
c (number of
cells)
5
6
7
8
9
10
11
12
...
1 + previous
value
In this example, use c = 5 because n = 24 is less than 25.
1.
Define the plot.
a.
From the Stats/List Editor, press „ Plots
(screen 10).
(10)
Note: From screen 10 you can
select 4:Fnoff if you have been
using function graphing. You could
also turn off StatPlots by selecting
3:PlotsOff.
b. Select 1:Plot Setup (screen 11).
(11)
Note: If you have plot information
displayed that you no longer need,
you can highlight it and deselect it
by pressing † Ÿ, or clear the line of
information by pressing … M.
© 2001 TEXAS INSTRUMENTS INCORPORATED
CHAPTER 1: GRAPHICAL DISPLAYS OF UNIVARIATE DATA
c.
27
With Plot 1 highlighted, press ƒ Define, and then
press B for Plot Types (screen 12).
(12)
d. Select 4:Histogram, paste phily in the x field, with
Hist. Bucket Width: 100, and
Use Freq and Categories?: NO (screen 13). (Bucket
width is calculated using
b ≈ (maxX – minX) P c = (945 – 400) P 5 = 109 ≈ 100.)
e.
Press ¸ to confirm the entry.
f.
Press ¸ to display the Plot Setup screen with
Plot 1 now defined and selected with a check (Ÿ) in
the left margin (screen 14).
(13)
(14)
Note: For all other plot types, press
‡ ZoomData. Set up the window
for histograms.
2.
Set up the window using ¥ $ with the following
entries:
•
xmin = 400 (because it is a nice number).
•
xmax = 1000 (approximately maxX = 945) (because it
works well with a bucket width of 100).
•
xscl = 0 (ensures that no tick marks will show).
•
ymin = -6 (-1/4 * n = -1/4 * 24 = -6) (to leave space below
the plot for output from trace or cursor coordinates).
•
ymax = 18 (3 * |ymin| = 3 * 6 = 18) (as a first estimate of
(15)
maximum cell height in the histogram).
•
yscl = 0
•
xres = 2
(See screen 15.)
© 2001 TEXAS INSTRUMENTS INCORPORATED
28
3.
ADVANCED PLACEMENT STATISTICS WITH THE TI-89
Graph the histogram. Press ¥ %, and then press …
Trace (screen 16).
The plot fits well on the screen. If the first cell (or any
other cell) had 20 values instead of 15, ymin and ymax
would have to be adjusted to leave approximately the
bottom quarter of the screen for output from the trace.
(16)
The histogram is best used with large data sets, however
screen 16 shows information similar to the stemplot and
dotplot of Topic 2 (data skewed to the right with an
obvious gap). The fact that you end up with six cells
(including the gap cell with frequency zero) while
c = 5, is of no concern since 5 was a starting
approximation and 24 is very close to 25.
4.
Construct a frequency table.
a.
From screen 16, notice that the first cell includes 15
values from 400 to 500. (The building that is 500 feet
tall goes in the second cell, not the first.)
b. Press B for the second cell. There are four values
from 500 to 600 (screen 17).
c.
Continue for the frequency table of Philadelphia
Tall Building Heights (in feet).
(17)
Philadelphia Tall Building Heights (in feet)
Cell limits
Cell midpoint
Frequency
400 to 500
450
15
500 to 600
550
4
600 to 700
650
0
700 to 800
750
3
800 to 900
850
1
900 to 1000
950
1
© 2001 TEXAS INSTRUMENTS INCORPORATED
CHAPTER 1: GRAPHICAL DISPLAYS OF UNIVARIATE DATA
5.
29
Change the bucket (or cell) width.
a.
Return to the Stats/List Editor using ¥ O.
b. Select Stats/List Editor and press ¸.
c.
Press „ Plots and select 1:Plot Setup (screen 14).
d. Press ƒ Define (screen 13).
e.
Press D twice and enter Hist. Bucket Width: 50.
Press ¸ to complete the entry.
f.
Press ¸ for an updated plot setup with the last
value now 50.
g.
Press N to return to the Stats/List Editor.
h. Press ¥ % for the graph screen, and then
press … Trace (screen 18).
In screen 9, you see three clusters and two gaps with a
possible outlier. The plot does not make the best use of the
screen space, and window values could be adjusted.
(18)
City Hall
One
Liberty
Place
Note: To return to the Stats/List
Editor from graph screen 18, press
2 a. This command toggles
between the current and previous
screens. Press ¥ O ¸ to
return to the Stats/List Editor.
Creating a Histogram from a Frequency Table
Example: Use the Philadelphia tall building height data
from the previous frequency table.
1.
2.
Return to the Stats/List Editor using ¥ O. Select
Stats/List Editor and press ¸.
Set up the Stats/List Editor with phily, list1, list2, and
list3.
a.
Highlight list1 and press M ¸.
b. Highlight list2 and press M ¸.
© 2001 TEXAS INSTRUMENTS INCORPORATED
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ADVANCED PLACEMENT STATISTICS WITH THE TI-89
c.
Highlight list3 and press M ¸ (screen 19).
d. Type the cell midpoints from the table of building
heights on page 28 in list1.
(19)
e.
Type the frequencies from the table on page 28 in
list2 (screen 20).
(20)
3.
Set up the plot.
a.
Press „ Plots. Select 1:Plot Setup to display the
Plot Setup screen.
b. Press ƒ Define to display the Define Plot 1 screen.
c.
Select Plot Type: Histogram, paste list1 in the x field
from the VAR-LINK screen, with
Hist. Bucket Width: 100, Use Freq and Categories?:
YES, and Freq: list2 (screen 21).
d. Press ¸ to confirm the entry.
e.
4.
Press ¸ to return to the Plot Setup screen and
N to return to the Stats/List Editor.
(21)
Set up the window using ¥ $ with the following
entries:
•
xmin = 400 (smallest cell limit from table)
•
xmax = 1000 (largest cell limit from table)
•
xscl = 0
•
ymin ≈ ë7 (-1/2 ∗ largest freq = -1/2 ∗ 15 ≈ -7)
•
ymax ≈ 21 (3 ∗ |ymin| = 3 ∗ 7 = 21)
•
yscl = 0
•
xres = 2
(See screen 22.)
© 2001 TEXAS INSTRUMENTS INCORPORATED
(22)
CHAPTER 1: GRAPHICAL DISPLAYS OF UNIVARIATE DATA
5.
31
Graph the histogram. Press ¥ %, and then press
… Trace (screen 23). This screen is similar to the
histogram from the raw data in screen 16. (The
windows differ a bit.)
(23)
Creating the Relative Frequency Histogram
The relative frequency histogram uses the previous example.
1.
Calculate the relative frequencies.
a.
Return to the Stats/List Editor using ¥ O.
b. Select Stats/List Editor and press ¸.
c.
Highlight list3.
d. Paste list2 (press 2 ° and select list2) and
divide by 24.0 (not 24), the sum of the frequencies
to get decimal results instead of fractions
(screen 24).
(24)
e.
Press ¸ (screen 25).
Observe that 62.5% of the buildings are 400 to 500 feet
tall and 16.7% are 500 to 600 feet tall.
If divided by 24 instead of 24.0, pressing ¥ ‘ (in
screen 25) also gives decimal results.
(25)
If proper fractions were obtained, you could (1)
highlight the list name, (2) press ¸ to highlight the
list in the input line, (3) press B to go to the end of the
set and multiply by a decimal ∗ 1.0, and (4) press ¸
to execute the operation. The resulting list will contain
decimals.
If you use the Mode screen and select Approximate, all
calculations would be decimal. Notice the AUTO or
APPROX designations in the middle of the bottom
status line.
© 2001 TEXAS INSTRUMENTS INCORPORATED
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ADVANCED PLACEMENT STATISTICS WITH THE TI-89
Graphing the Relative Frequency Histogram
To set up the relative frequency histogram, you need to
return to the Define Plot 1 screen and change the Freq: field
to list3, the relative frequencies.
1.
Press „ Plots, 1:Plot Setup, and then press ƒ Define.
2.
Replace Freq: list2 with list3 (screen 26).
3.
Change the window so that ymax and ymin are based
on the greatest value, .625.
(26)
4.
Set up the window using ¥ $ with the following
entries:
•
xmin = 400
•
xmax = 1000
•
xscl = 0
•
ymin ≈ ë0.3 (-1/2 ∗ largest value in y
•
ymax ≈ 0.9 (3 ∗ |ymin| = 3 ∗ 0.3 = 0.9)
•
yscl = 0
•
xres = 2
(27)
list = -1/2 ∗ 0.625 ≈ -0.3)
(See screen 27.)
5.
Press ¥ %, and then press … Trace (screen 28).
This histogram is similar in shape to screen 23, but now
n = 0.625 = 62.5% instead of 15. This n-value represents a
relative frequency (15/24) instead of the frequency (15).
(28)
© 2001 TEXAS INSTRUMENTS INCORPORATED
CHAPTER 1: GRAPHICAL DISPLAYS OF UNIVARIATE DATA
33
Topic 4—Frequency Plots
Cumulative Frequency Plots
Example: Use the building heights data in list phily.
To develop a cumulative frequency plot, you need the
frequency list of the Define Plot 1 screen to be the list of
cumulative frequencies.
1.
Calculate the cumulative frequencies.
a.
Return to the Stats/List Editor using ¥ O.
b. Select Stats/List Editor and press ¸.
c.
Highlight list1 and press M ¸. Do the same
for list2 and list3.
d. Enter the lower cell limits from the frequency table
in Topic 3 in list1. Enter the last value of 1000, the
upper limit of the last cell.
e.
2.
Enter the frequencies from the table in Topic 3 in
list2. Insert a 0 as the first value, indicating zero
values below 400 feet, then 15 to indicate the
frequency of the class with the upper limit of 500
feet, and so forth (screen 29).
Create the list of cumulative frequencies.
a.
(29)
Highlight list3.
b. Press … List.
c.
Select 2:Ops and then 6:cumSum.
d. To complete the command, paste or type list2 and
close the parenthesis in the entry line (screen 30).
(30)
© 2001 TEXAS INSTRUMENTS INCORPORATED
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ADVANCED PLACEMENT STATISTICS WITH THE TI-89
e.
3.
Press ¸ to display the values in list3
(screen 31).
Set up the plot and graph the cumulative frequency
plot.
a.
Press „ Plots. Select 1:Plot Setup to display the
Plot Setup screen.
b. Press ƒ Define to display the Define Plot 1 screen.
c.
(31)
Note: The third value in list3 is the
sum of the first 3 rows of list2 or
0 + 15 + 4 = 19, indicating 19
values less than 600 feet tall. The
last value is the sum of all the
frequencies of list2 or n = 24 (not
shown).
Select Plot Type: xyline, Mark: Box, x: list1, y: list3,
and Use Freq and Categories?: NO (screen 32).
d. Press ¸ to return to the Plot Setup screen.
(32)
e.
Press ‡ ZoomData, and then press … Trace and B
B (screen 33).
The value traced in screen 33 shows that 19
buildings (yc = 19) have heights below 600 feet
(xc = 600).
Interpretation of the plot: The plot starts with a steep slope,
indicating many of the buildings are at the smaller height.
Cumulative Relative Frequency Plot
This plot requires a list of cumulative relative frequencies.
1.
Calculate the cumulative relative frequencies.
a.
Return to the Stats/List Editor using ¥ O.
b. Select Stats/List Editor and press ¸.
c.
Highlight list4 (screen 31).
d. Paste list3, and then divide by 24.0 (do not forget
the decimal point). (See screen 31.)
© 2001 TEXAS INSTRUMENTS INCORPORATED
(33)
CHAPTER 1: GRAPHICAL DISPLAYS OF UNIVARIATE DATA
e.
35
Press ¸ (screen 34).
Observe that 0.79167 in the third row indicates that
79.2% of the buildings (19/24) are less than 600 feet tall.
The last value in the list of 1.0 (not shown) indicates
that all of the buildings (100%) are less than 1000 feet tall.
2.
Set up the plot and graph the relative cumulative
frequency.
a.
Press „ Plots. Select 1:Plot Setup to display the
Plot Setup screen.
b. Press ƒ Define to display the Define Plot 1 screen.
c.
(34)
Note: The order of operations in
screens 29 and 30 could be
reversed. The relative frequencies
could have been found (as in Topic
3, screen 25, list3), and then the
cumulative sum found for list4
(screen 34).
Select Plot Type: xyline, Mark: Box, x: list1, y: list4,
and Use Freq and Categories?: NO (screen 35).
(35)
d. From the Plot Setup screen, press ‡ ZoomData,
and then press … Trace and B B (screen 36). This
plot is the same shape as screen 33, but now it has
cumulative relative frequencies.
(36)
© 2001 TEXAS INSTRUMENTS INCORPORATED